Why Negative State Prices Conflict with Monotonic Utility

Summary

The document asks whether negative state prices, which may appear when extracting risk-neutral distributions from option prices, can have an economic interpretation. It describes a state price as the current value of a claim paying one unit in a particular state and nothing in other states. The question relates those prices to equilibrium and utility, where agents trade to smooth consumption according to their preferences.

The answer argues that a negative state price is incompatible with utility that is strictly increasing in wealth across all states: an agent who always prefers more wealth would not pay to receive a claim that can only add wealth in one state, yet has a negative price. The argument depends on that preference assumption. The document does not examine alternative preferences, market frictions, or practical causes of negative values in option-based estimates, so its conclusion is theoretical and conditional rather than a full treatment of observed estimates.

Key ideas

• A state price values a payoff that occurs in one specified state of the world.
• With strictly increasing utility, agents prefer more wealth to less in every state.
• Under that assumption, equilibrium state prices cannot be negative.
• The argument does not address alternative preferences or estimation issues in option data.

Tags

# Negative risk neutral probabilities economic argument


# Negative risk neutral probabilities economic argument












We know of plenty ways to extract risk neutral distirbutions from option prices (for example Breeden Litzberger) but there is no real analysis on how to interpret negative state prices (Haug 2007 for example). State prices are Arrow Debreu securities, so it is the price an agent is willing to pay to get $1\$$ in a particular state and $0$ else.

Doing equilibirum and utility maximization we know that the agents smoothen their consumption by marginal rate of substitution which describes their risk-aversion.

Coming back to the negative probabilties: Could it not be possible there exist actual economic states where the agent is receiving money for the contract?

## Answer by pbr142 (score 1)

https://quant.stackexchange.com/a/21889

Negative state prices in equilibrium can only occur if the utility of wealth is not always strictly positive. A natural assumption on utility is that it is strictly increasing in wealth in all states of nature (more money is always preferred to less money). Thus, unless you can come up with a situation in which an agent would prefer to be less wealthy and everything else equal there should not be negative state prices.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.